THE MOTION OF A SPHERE 191 



Putting v = -j 1 - and integrating again with condition y = when t = o we get 



y=v\t-\{l-e-«)\ (6) 



•_'. Horizontal motion with initial velocity H. 



If u is horizontal velocity at time t, the equation of motion is now simply 



du 

 di = - cu > 



or 



du 

 u -7-= -cu. 

 dx 



Therefore 



du = - cdx, 

 and hence 



K-u = cx (7) 



But .'■ = X for u = 0, therefore H = cX. 



From the last expression and (3) we obtain 



«=f (8) 



Proceeding with the integration, from (7) we have 



dx 

 -dt = U 



= U-cx 



= e(X-x). 



Integration with initial condition x = when t = o leads to 



*=X(1-*-*) (9) 



3. The equation of the path of a sphere projected horizontally under gravity 

 is obtained at once by the elimination of t from the two equations (6) and (9) ; 

 and replacing c by its value ? we have finally 



»-7i- to «'( 1 -5)-i} < 10 » 



